The dawn of mathematical biology
A aurora da biologia matemática
Daniel Sander Hoffmann
Universidade Estadual do Rio Grande do Sul
[email protected]
http://lattes.cnpq.br/4388320562511213
Abstract
In this paper I describe the early development of the so-called mathematical biophysics, as conceived by
Nicolas Rashevsky back in the 1920´s, as well as his latter idealization of a “relational biology”. I also
underline that the creation of the journal The Bulletin of Mathematical Biophysics was instrumental in
legitimating the efforts of Rashevsky and his students, and I finally argue that his pioneering efforts,
while still largely unacknowledged, were vital for the development of important scientific contributions,
most notably the McCulloch-Pitts model of neural networks.
Keywords
Mathematical biophysics; Neural networks; Relational biology.
Resumo
Neste artigo eu descrevo o desenvolvimento inicial da assim chamada biofísica matemática, da forma
como esta foi concebida por Nicolas Rashevsky desde a década de 1920, bem como a sua posterior
idealização de uma “biologia relacional”. Eu também enfatizo que a criação do periódico The Bulletin of
Mathematical Biophysics foi instrumental na legitimação dos esforços de Rashevsky e de seus estudantes, e
finalmente argumento que os seus esforços pioneiros, embora ainda largamente desconsiderados, foram
vitais para o desenvolvimento de importantes contribuições científicas, destacando-se o modelo de redes
neurais de McCulloch-Pitts.
Palavras-chave
Biofísica matemática; Biologia relacional; Redes neurais.
1. Introduction
The modern era of theoretical biology can be classified into “foundations”, “physics and
chemistry”, “cybernetics”, and “mathematical biophysics” (Morowitz, 1965). According to this
author, an important part of the history of the modern era in theoretical biology dates back to
the beginning of the twentieth century, with the publication of D’Arcy Wentworth Thompson’s
opus On Growth and Form (Thompson, 1917), closely followed by works like the Elements of
Physical Biology of Alfred J. Lotka (1925). Some authors tracked Lotka’s ideas closely, yielding
books such as Leçons sur la Théorie Mathématique de la Lutte pour la Vie by Vito Volterra (1931),
and Kostitzin’s Biologie Mathématique (Kostitzin, 1937). Mathematics and ecology do share a long
coexistence, and mathematical ecology is currently one of the most developed areas of the
theoretical sciences taken as a whole. Genetics is another example of great success in modern
applied mathematics, its history beginning (at least) as early as in the second decade of the last
century, when J. B. S. Haldane published A Mathematical Theory of Natural and Artificial Selection
(Haldane, 1924), followed by The Genetical Theory of Natural Selection in 1930, and The Theory of
Inbreeding in 1949, both by R. A. Fisher (see Fisher, 1930; 1949). The approach of “physics and
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53
chemistry” is represented by workers like Erwin Schrödinger – one of the founding fathers of
quantum mechanics –, who wrote a small but widely read book entitled What is Life?
(Schrödinger, 1944), and Hinshelwood (1946), with his The Chemical Kinetics of the Bacterial Cell.
Cybernetics is a successful term coined by Norbert Wiener and used in a book with the same
name (Wiener, 1948). At the same time, C. E. Shannon (1948) published his seminal paper A
Mathematical Theory of Communication and it is a known fact that both works profoundly
influenced a whole generation of mathematical biologists and other theoreticians. While Wiener
stressed the importance of feedback – with the notion of closed loop control yielding new
approaches to theoretical biology, ecology and the neurosciences –, information theory was
improved and applied in several technological and scientific areas. Finally, the amalgamation of
information theory with the notion of feedback strongly influenced the work of important
theoretical ecologists like Robert E. Ulanowicz (1980; 1997) and Howard T. Odum (1983).
I suggest that the development of the last division highlighted above, namely
“mathematical biophysics” is, up to now, largely unknown to mainstream historians and
philosophers of science. Interestingly enough, this unfamiliarity spreads even to most historians
and philosophers of biology. However, I wish to point out a recent revival of some fundamental
ideas associated with this school, a fact that, alone, justifies a closer look into the origins of this
investigative framework. Accordingly, in this paper I review and briefly discuss some early
stages of this line of thought.
2. The roots of mathematical biophysics and of the relational approach
Nicolas Rashevsky was born in Chernigov on September 1899 (Cull, 2007). He took a Ph.D. in
theoretical physics very early in his life, and soon began publishing in quantum theory and
relativity, among other topics. He immigrated to North America in 1927, after being trained in
Russia as a mathematical physicist. His original work in biology began when he moved to the
Research Laboratories of the Westinghouse Corporation in Pittsburgh, Pennsylvania, where he
worked on the thermodynamics of liquid droplets. There he found that these structures became
unstable past a given critical size, spontaneously dividing into smaller droplets. Later still,
while involved with the Mathematical Biology program of the University of Chicago,
Rashevsky studied cell division and excitability phenomena. The Chicago group established The
Bulletin of Mathematical Biophysics (now The Bulletin of Mathematical Biology), an important
contribution to the field of theoretical biology. In this journal one finds most of Rashevsky’s
published biological material, and it also served to introduce the work of many of his students.
Thus, the journal helped to catapult new careers and (above all) catalyze the formation and
maintenance of the “mathematical biophysics” school, still an influential school in modern
theoretical biology (Rosen, 1991). To be fair, the journal was widely open to all interested
researchers. More than that, Rashevsky and his colleague Herb Landahl used to take for
themselves the task of correcting and even helping to extend the mathematics, encouraging the
authors to re-submit the papers. As a side note, it is interesting to mention that, given the
shortcomings of the publication of graphics at the time, Landahl offered invaluable help to the
authors, carefully preparing each drawing for printing (Cull, 2007).
The importance of organizing new journals, proceedings and books for “legitimating” a
new branch of science was emphasized by Smocovitis (1996). I would like to suggest, therefore,
that the “mathematical biophysics” case fits nicely this interpretation. Other periodicals of
importance that arose during this period include Acta Biotheoretica founded in 1935 by the group
then at the Professor Jan der Hoeven Foundation for Theoretical Biology of the University of
Leiden, Bibliographia Biotheoretica (published by the same group) and the well-known Journal of
Theoretical Biology, founded in 1961 (Morowitz, 1965).
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Rashevsky is rightly acknowledged for the proposition of a systematic approach to the
use of mathematical methods in biology. He intended to develop a “mathematical biology” that
would relate to experiments just like the well-established mathematical physics (Cull, 2007). He
chose to name this new field of inquiry “mathematical biophysics”, a decision reflected in the
title of the aforementioned journal. By the mid-1930s Rashevsky had already explored the links
between chemical reactions and physical diffusion (currently known as “reaction-diffusion”
phenomena), as well as the associated destabilization of homogeneous states that is at the core
of the modern notion of self-organization (Rosen, 1991). An elaborated theory of cell division
based on the principles behind diffusion drag forces was offered close to the end of that decade
(Rashevsky, 1939), and led to new equations concerning the rates of constriction and elongation
of demembranated Arbacia eggs under division (Landahl, 1942a; 1942b). (I note that Arbacia is a
genus of hemispherically-shaped sea urchins commonly used in experiments of the kind.) This
theory agreed well with previously available empirical data, but Rashevsky himself urged to
demonstrate that the theory of diffusion drag forces was inadequate to represent most other
facts of cell division known at the time (Rosen, 1991).
Rashevsky’s major work is, rather unsurprisingly, entitled Mathematical Biophysics
(Rashevsky, 1960), a book that was revised and reedited more than once (most of the above
mentioned studies can be found in this publication). The original edition, dating back to 1938,
covered cellular biophysics, excitation phenomena and the central nervous system, with
emphasis in the physical representation (Morowitz, 1965). Subsequently, Rashevsky delved into
still more abstract mathematical approaches, as I describe later.
I wish to point out that the initial efforts of Rashevsky are both important and largely
unrecognized by contemporary philosophers and historians of science. Indeed, the academic
infrastructure and the research agenda established by this forerunner were crucial in
subsidizing the development of important scientific contributions made by other researchers.
Perhaps the most unexpected case is the crucial role played by Rashevsky in the line of
investigation leading to the celebrated McCulloch-Pitts model of neural networks, published in
1943. Consider the following statement, which nicely summarizes a commonly held belief of the
scientific and philosophical communities: “The neural nets branch of AI began with a very early
paper by Warren McCulloch and Walter Pitts...” (Franklin, forthcoming). Actually, the first
mathematical descriptions of the behavior of “nerves” and networks of nerves are to be credited
to Rashevsky, who during the early 1930´s published several papers concerning a mathematical
theory of conduction in nerves, based on electrochemical gradients and the diffusion of
substances (Abraham, 2002). Rashevsky´s fundamental idea was to use two linear differential
equations together with a nonlinear threshold operator (Rashevsky, 1933). It was in this paper,
hence, that Rashevsky's “two-factor” theory of nerve excitation became public for the first time.
This theory was based on the diffusion kinetics of excitors and inhibitors (Abraham, 2002). Only
many years later the unknown “substances” were correctly identified as concentrations of
sodium and potassium, thanks to the important work of Hodgkin and Huxley (Cull, 2007). I am
not willing to get into the technical details here (a thorough investigation will be published
elsewhere), but suffice it to say that Rashevsky argued that his simple mathematical model
could fit empirical data, available at the time, regarding the behavior of single neurons. Still
more important, he postulated that these model neurons could be connected in networks, in
order to yield complex behavior, and even allow the modeling of the entire human brain (Cull,
2007). Latter, close to the end of that decade, Walter Pitts was introduced to Rashevsky by
Rudolf Carnap, and accepted into his mathematical biology group (Cowan, 1998). Together,
Pitts (who was a superb mathematician) and the “philosophical psychiatrist” Warren S.
McCulloch (Abraham, 2002) published their groundbreaking paper, entitled A Logical Calculus
of the Ideas Immanent in Nervous Activity (McCulloch and Pitts, 1943). Many years later,
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McCulloch recalled that he and Pitts were able to publish their ideas in Rashevsky´s journal
thanks to Rashevsky´s defense of mathematical and logical ideas in the field of biology
(Abraham, 2002). The objective fact that Rashevsky was apparently the first investigator to
come up with the idea of a “neural net” mathematical model (Rosen, 1991), however, was
largely neglected.
According to Rosen (1991), by the 1950s Rashevsky had explored many areas of
theoretical biology, but he felt that his approach still lacked “genuinely fresh insights”. Thus, he
suddenly took a wholly new research direction, turning from mathematical methods closely
associated with empirical data to an overarching search for general biological principles.
Putting aside Rosen´s opinion, this radical turn is seemingly most easily justifiable simply as a
strong reaction to Rashevsky´s own critics, who claimed that his mathematical biophysics
approach was not a novelty anymore (ironically) –, given that many researchers had already
began incorporating models derived from physics, as well as quantitative methods, in their
work (Cull, 2007).
Anyway, the fact is that Rashevsky expressed a bunch of novel ideas in a paper
interestingly entitled Topology and Life: In Search of General Mathematical Principles in Biology and
Sociology. In the paper, after pointing out most major developments in the mathematical biology
of the time, he goes on saying: “All [these] theories... deal with separate biological phenomena.
There is no record of a successful mathematical theory which would treat the integrated
activities of the organism as a whole” (Rashevsky, 1954, p. 319-320). According to him, it was
important to have the knowledge that diffusion drag forces are responsible for cell division and
that pressure waves are reflected in blood vessels, as well as to have a mathematical formalism
for dealing with complicated neural networks. But then he emphasized that there was nothing
so far in these theories indicating that an adequate functioning of the circulatory system was
fundamental for the normal operation of intracellular processes; Furthermore, there was
nothing in the formalisms showing that an elaborated process in the brain, that resulted, e.g., in
the location of food, was causally connected with metabolic processes going on in the cells of
the digestive system. The same was true regarding the causal nexus between a failure in the
normal behavior of a network of neurons and the cell divisions that resulted from a stimulation
of the process of healing due to the accidental cutting of, say, a thumb (Rosen, 1991). Yet,
according to him, “this integrated activity of the organism is probably the most essential
manifestation of life” (Rashevsky, 1954, p. 320). Unfortunately, Rashevsky argued, one usually
approaches the effects of these diffusion drag forces simply as a diffusion problem in a
specialized physical system, and one deals with the processes of circulation simply as special
hydrodynamic problems. Hence, the “fundamental manifestations of life” are definitely
excluded from all those biomathematical theories. In other words, biomathematics, according to
Rashevsky, lacked a capacity to adequately describe true integration of the parts of any organic
system. As a result, it was useless to try to apply the physical principles, used in the aforesaid
mechanical models of biological phenomena, to develop a comprehensive theory of life. Similar
lines of criticism could be applied, I submit, to modern theoretical frameworks attempting to
provide integrated models of the living organism (or the brain itself). This line of thought was
further developed by Rashevsky´s student Robert Rosen, who yielded an interesting theoretical
framework, built around a special notion of complexity. He also promoted the use of new
mathematical tools, like category theory, in theoretical biology investigations (Rosen, 1991;
2000).
According to Rashevsky, putting aside the possibility of constructing a
physicomathematical theory of the organism based on the physicochemical dynamics of cells and of
cellular aggregates does not prevent one from trying to find alternative pathways. What are the
possibilities, then? The key to understanding Rashevsky’s perspective, I suggest, is to start
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analyzing some of his fundamental premises. In fact, Rashevsky used to believe that the
biomathematics of his time was in a pre-Newtonian stage of development, despite the elaborate
theories then available. In pre-Newtonian physics there existed simple mathematical treatments
of isolated phenomena, but it was only with the arrival of Newton´s principles, incorporated in
his laws of motion, that physics attained a more comprehensive and unified synthesis. Ordinary
models of biomathematics, like the models of theoretical physics, are all based in physical
principles. But Rashevsky seemed to suggest that they should instead be based on genuinely
biological principles, in order to capture the integrated activities of the organism as a whole (Rosen,
1991). This is apparent in his words: “We must look for a principle which connects the different
physical phenomena involved and expresses the biological unity of the organism and of the
organic world as a whole” (Rashevsky, 1954, p. 321 – italics added). He also argued that
mathematical models are transient in nature, while a general principle, when discovered, is
perennial. For example, it is possible to devise several alternative models, all obeying the laws
of Newton (one model being e.g. the “billiard ball” molecule of the kinetic theory of gases). On
the same guise, there are distinct cosmological models, all based upon Einstein’s fundamental
principles (Peacock, 1999; Dalarsson and Dalarsson, 2005). It is clear then that Rashevsky
wished to conceive general principles, in biology, enjoying the same status earned by principles
in theoretical physics. After all, he was trained as a theoretical physicist.
3. Looking for general biological principles
I already highlighted the fact that Rashevsky was a researcher that eagerly pursued general
biological principles, and he indeed explicitly managed to propose some. In what follows, I
briefly examine (following Rosen, 1991) the nature of some of these principles. The first
principle I would like to emphasize is Rashevsky’s principle of adequate design of organisms,
originally denominated principle of maximum simplicity, and introduced as early as in 1943. As
originally formulated it states that, given that the same biological functions can be performed
by different structures, the particular structure found in nature is the simplest one compatible
with the performance of a function or set of functions. The principle of maximum simplicity
applies therefore to different models of mechanisms, of which the simplest one is to be preferred.
But given that simplicity is a vague notion in this case, being difficult to find out a measurement
standard, Rashevsky self-critically turned to a slightly different version, denominated principle
of optimal design. In this case, it is required that a structure necessary for performing a given
function be optimal relatively to energy and material needs. But it can be argued that there is still
some imprecision here, because a structure that is optimal with respect to material needs is not
necessarily optimal as far as energy expenditures are concerned. Hence, a more straightforward
notion was in need. Accordingly, Rashevsky turned to the last formulation of his principle, this
time putting aside the notion of optimality:
When a set of functions of an organism or of a single organ is prescribed, then, in order to
find the shape and structure of the organ, the mathematical biologist must proceed just as
an engineer proceeds in designing a structure or a machine for the performance of a given
function. The design must be adequate to the performance of the prescribed function under
specified varying environmental conditions. This may be called the principle of adequate
design of the organism (Rashevsky, 1965, p. 41 – italics added).
I note that the notion of “adequate” is still also a bit vague, but I am not pursuing this
discussion further in this paper. Let me instead raise a more interesting question, which
repeatedly arises in philosophy of biology, particularly in those areas of inquiry closely
associated to the Darwinian theory of biological evolution: is Rashevsky´s principle of adequate
design of the organism teleological? To this objection, Rashevsky himself had an answer: all
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variational principles in physics are “teleological” or “goal-directed”, beginning with the
principle of least action (see Lanczos, 1970, for a detailed mathematical account of this and other
physical principles). Other investigators subsequently offered similar justifications, e.g. Robert
Rosen, who dedicated a full chapter of his book Essays on Life Itself (Rosen, 2000), entitled
Optimality in Biology and Medicine, to this technical discussion. Another objection that Rashevsky
was well aware of was that the principle of adequate “design” seemed to imply some sort of
creative intelligence. Must one, in this case, presuppose a “universal engineer” of sorts? Not
necessarily, because, like most scientific principles, the principle of adequate design of the organism
“offers us merely an operational prescription for the determination of organic form by
calculation” (Rashevsky, 1965, p. 45). Here, I suggest that one would perhaps do best simply not
employing the term “design”, that also seems to be rather misguiding in this context. On the
other hand, according to Rashevsky, the principle could perhaps follow directly from the
operation of natural selection, which would only preserve “adequate” organisms, although it
could turn out to be an independent principle. This too, I suggest, could be nourishment for
heated discussions among contemporary philosophers of biology.
Perhaps still more thought-provoking is Rashevsky´s “principle of biological
epimorphism”, that emphasizes qualitative relations as opposed to quantitative aspects,
topology instead of metrics. It can be argued that a given biological property in a higher
organism has many more elementary processes than the equivalent biological property of a
lower one. Examples of biological properties are perception, locomotion, metabolism, etc. The
principle is based upon the fact that different organisms can be epimorphically mapped onto
each other, after the biological properties were already clearly distinguished and represented. In
such epimorphic mappings, the basic relations characterizing the organism as a whole are
preserved. Given Rashevsky’s mathematical proclivities, he wanted to put his principle into a
precise and rational context. Among the several branches of relational mathematics, topology
reigns supreme. Before going on, I think it necessary to briefly digress about this topic.
It is a known fact that topological ideas are present in most branches of modern
mathematics. In a nutshell, topology is the mathematical study of properties of objects, which
are preserved through deformation, stretching and twisting (tearing is forbidden). Hence, one is
entitled to say that a circle is topologically equivalent to an ellipse, given that one can be
transformed into the other by stretching. The same is valid for a sphere, which can be
transformed into an ellipsoid, and vice versa. Topology has indeed to do with the study of
objects like curves, surfaces, the space-time of Minkowsky (in relativity theory, see Peacock,
1999), physical phase spaces and so on. Furthermore, the objects of topology can be formally
defined as “topological spaces”. Two such objects are homeomorphic if they have the same
topological properties. Using such perspective, Rashevsky postulated that to each organism
there is a corresponding topological “complex”. More complicated complexes correspond to
higher organisms, and different complexes are converted into each other by means of a
universal rule of geometrical transformation. Furthermore, they can be mapped onto each other
in a many-to-one manner, preserving certain basic relations. Rashevsky expressed his “principle
of biological epimorphism” by postulating that, if one represents geometrically the relations
between several functions of an organism in a single convenient topological complex, then the
topological complexes that represent different organisms are obtainable, via a proper
transformation, from just one or a few primordial topological complexes. A previous version of
this principle is what Rashevsky used to call the “principle of bio-topological mapping”.
According to this principle, the topological complexes by means of which diverse organisms are
represented are all obtainable from one or a few primordial complexes by the same
transformation. This transformation contains one or more parameters, different values of it
corresponding to different organisms (Rashevsky, 1954). The considerations above may
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hopefully give us a glimpse of Rashevsky’s relational approach to the study of life, epitomized
in the expression “relational biology”, that he coined in order to help delineate a clear
framework for thinking in the life sciences.
4. Conclusion
Nicolas Rashevsky (who passed away in 1972) was a pioneer in theoretical biology, having
inaugurated the school of “mathematical biophysics” and subsequently pioneered the field of
“relational biology” or (still another term that he coined) “biotopology”. I call attention to the
fact that the latter must definitely be distinguished from “topobiology”, a term coined by Nobel
laureate Gerald Edelman in the context of cell and embryonic development research. Edelman´s
theory postulates that differential adhesive interactions among heterogeneous cell populations
drive morphogenesis, and explains, among other things, how a complex multi-cellular
organism can arise from a single cell (Edelman, 1988).
I already emphasized that the creation of The Bulletin of Mathematical Biophysics was an
important tool for establishing and helping broadcast Rashevsky´s work (as well as the
proposals of his own students). Furthermore, I pointed out that the work of Rashevsky implies
that at least some aspects of contemporary theoretical biology and neuroscience have older
roots than previously thought. This is exemplified by Rashevsky´s active role in the body of
research that paved the way to the development and publication of the McCulloch-Pitts model
of neural networks. Finally, I suggested that Rashevsky´s criticism of purely mechanical and
non-integrative approaches to biology may as well be evaluated under the light of current
theories, including proposals in theoretical biology – and, once again, in neuroscience. The
critical analysis of these claims, I submit, is an interesting and yet largely unexplored subjectmatter to philosophers of science.
Rashevsky´s influence still reverberates in important scientific research areas such as
neural networks and non-equilibrium pattern formation, among others. However, as I see it,
relational biology effectively came of age with the far more encompassing and methodical work
of Rashevsky’s former student Robert Rosen (who passed away in 1998 – see Rosen 1991; 2000)
and of his followers. A very active contemporary player worth mentioning is a bright pupil of
Rosen, the mathematical biologist Aloisius H. Louie (see Louie, 2009; 2013).
Acknowledgments
Some ideas presented in this paper first came to my mind during the development of my Ph.D.
dissertation, years ago, and ended up as part of a chapter (still unpublished). Accordingly, I
would like to thank CNPq for the support at the time, by means of a much needed research
grant.
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